Discovering a Formula for the Difference of Squares in the Fibonacci Sequence

[06Rousher]

Problem is: "By experimenting with numerous examples in search of a pattern, determine a simple formula for (F n+1)^2-(F n-1)^2; That is, a formula for the difference of the squares of two Fibonacci numbers."

The n+1 and n-1 should be smaller by the F but I don't know how to do that on a computer

Any help is appreciated

 

[MarkFL]

We are asked to find a formula for:

\(\displaystyle A_{n}=F_{n+1}^2-F_{n-1}^2\)

So, as suggested, let's see if a pattern develops:

\(\displaystyle A_1=F_2^2-F_0^2=1^2-0^2=1=F_2\)

\(\displaystyle A_2=F_3^2-F_1^2=2^2-1^2=3=F_4\)

\(\displaystyle A_3=F_4^2-F_2^2=3^2-1^2=8=F_6\)

\(\displaystyle A_4=F_5^2-F_3^2=5^2-2^2=21=F_8\)

At this point, we could state the induction hypothesis $P_n$:

\(\displaystyle F_{n+1}^2-F_{n-1}^2=F_{2n}\)

Can you proceed?

 

[06Rousher]

Proceed with continuing the pattern?

Im not understanding the hypothesis of F 2n aswell

 

[MarkFL]

Proceed with continuing the pattern?

Im not understanding the hypothesis of F 2n aswell

I mean can you continue the proof by induction. The hypothesis is what we notice appears to be the pattern that arises when computing the first several terms of the sequence we are asked to explore. Have you been using induction in your course?

 

[06Rousher]

No i have no clue on induction

Im helping a friend with his work and trying to understand it myself cause I know it will be in my future. So I haven't had guidelines or someone to teach me, just been trying to do this on my own